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Answer :
Let's go through the process of subtracting the given rational expressions step by step to identify the mistake Morgan made.
### Problem:
You have the rational expressions:
1. [tex]\(\frac{3t^2 - 4t + 1}{t+3}\)[/tex]
2. [tex]\(\frac{t^2 + 2t + 2}{t+3}\)[/tex]
And the subtraction:
[tex]\[
\frac{3t^2 - 4t + 1}{t+3} - \frac{t^2 + 2t + 2}{t+3}
\][/tex]
### Step-by-step Solution:
1. Align the Denominators:
Both expressions have the same denominator, [tex]\(t + 3\)[/tex], so we can directly subtract the numerators.
2. Subtract the Numerators:
[tex]\[
(3t^2 - 4t + 1) - (t^2 + 2t + 2)
\][/tex]
3. Distribute the Negative Sign:
Subtracting the second expression means distributing the negative sign to each term inside the parentheses:
[tex]\[
- (t^2 + 2t + 2) = -t^2 - 2t - 2
\][/tex]
4. Combine the Terms:
Now combine all like terms from the two numerators:
[tex]\[
3t^2 - 4t + 1 + (-t^2 - 2t - 2)
\][/tex]
5. Perform the Arithmetic on the Numerators:
- [tex]\(3t^2 - t^2 = 2t^2\)[/tex]
- [tex]\(-4t - 2t = -6t\)[/tex]
- [tex]\(1 - 2 = -1\)[/tex]
So the combined numerator becomes:
[tex]\[
2t^2 - 6t - 1
\][/tex]
6. Identify the Error:
Morgan's result was given as [tex]\(\frac{2t^2 - 2t + 3}{t+3}\)[/tex]. Instead, it should have been [tex]\(\frac{2t^2 - 6t - 1}{t+3}\)[/tex] based on the correct steps.
### Conclusion:
The error Morgan made was "forgetting to distribute the negative sign to two of the terms in the second expression." Properly distributing the negative sign is crucial in subtracting rational expressions to ensure all terms are correctly subtracted.
### Problem:
You have the rational expressions:
1. [tex]\(\frac{3t^2 - 4t + 1}{t+3}\)[/tex]
2. [tex]\(\frac{t^2 + 2t + 2}{t+3}\)[/tex]
And the subtraction:
[tex]\[
\frac{3t^2 - 4t + 1}{t+3} - \frac{t^2 + 2t + 2}{t+3}
\][/tex]
### Step-by-step Solution:
1. Align the Denominators:
Both expressions have the same denominator, [tex]\(t + 3\)[/tex], so we can directly subtract the numerators.
2. Subtract the Numerators:
[tex]\[
(3t^2 - 4t + 1) - (t^2 + 2t + 2)
\][/tex]
3. Distribute the Negative Sign:
Subtracting the second expression means distributing the negative sign to each term inside the parentheses:
[tex]\[
- (t^2 + 2t + 2) = -t^2 - 2t - 2
\][/tex]
4. Combine the Terms:
Now combine all like terms from the two numerators:
[tex]\[
3t^2 - 4t + 1 + (-t^2 - 2t - 2)
\][/tex]
5. Perform the Arithmetic on the Numerators:
- [tex]\(3t^2 - t^2 = 2t^2\)[/tex]
- [tex]\(-4t - 2t = -6t\)[/tex]
- [tex]\(1 - 2 = -1\)[/tex]
So the combined numerator becomes:
[tex]\[
2t^2 - 6t - 1
\][/tex]
6. Identify the Error:
Morgan's result was given as [tex]\(\frac{2t^2 - 2t + 3}{t+3}\)[/tex]. Instead, it should have been [tex]\(\frac{2t^2 - 6t - 1}{t+3}\)[/tex] based on the correct steps.
### Conclusion:
The error Morgan made was "forgetting to distribute the negative sign to two of the terms in the second expression." Properly distributing the negative sign is crucial in subtracting rational expressions to ensure all terms are correctly subtracted.
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